The Min Cost Flow Problem

1. The Min Cost Flow Problem

2. The Min Cost Flow problem • We want to talk about multi-source, multi-sink flows than just “flows from s to t”. • We want to impose lower bounds as well as capacities on a given arc. Also, arcs should have costs. • Rather than maximize the value (i.e. amount) of the flow through the network we want to minimize the cost of the flow.

3. Flow Networks with Costs • Flow networkswith costs are the problem instances of the min cost flow problem. • A flow network with costs is given by 1) a directed graph G = (V,E) 2) capacitiesc: E!R. 3) balances b: V!R. 4) costsk: V£V !R. k(u,v) = -k(v,u). • Convention: c(u,v)=0 for (u,v) not in E.

4. Feasible Flows • Given a flow network with costs, a feasible flow is a feasible solution to the min cost flow problem. • A feasible flow is a map f: V £V!R satisfying capacity constraints: 8 (u,v): f(u,v) ·c(u,v). Skew symmetry: 8 (u,v): f(u,v) = – f(v,u). Balance Constraints: 8u2V: v2Vf(u,v) =b(u)

5. Cost of Feasible Flows • The cost of a feasible flow f is cost(f)= ½ (u,v) 2 V £V k(u,v) f(u,v) • The Min Cost Flow Problem: Given a flow network with costs, find the feasible flow f that minimizes cost(f).

6. Max Flow Problem vs. Min Cost Flow Problem Max Flow Problem Problem Instance: c: E!R+. Special vertices s,t. Feasible solution: 8u2V–{s,t}: v2Vf(u,v) = 0 Objective: Maximize |f(s,V)| Min Cost Flow Problem Problem Instance: c: E!R. Maps b,k. Feasible solution: 8u2V: v2Vf(u,v) = b(u) Objective: Minimize cost(f)

7. Negative Capacities • c(u,v) < 0. • Let l(v,u) = – c(u,v). • f(u,v) ·c(u,v) iff f(v,u) ¸ – c(u,v) = l(v,u). • l(v,u) is a lower bound on the net flow from v to u for any feasible flow.

8. Balance Constraints • Vertices u with b(u)>0 are producing flow. • Vertices u with b(u)<0 are consuming flow • Vertices u with b(u)=0 are shipping flow.

9. . Unknown Balance!! Unknown Balance!! Can we solve the max-flow problem using software for the min-cost flow problem?

10. All other costs 0, all balances 0. Capacity 29, Cost -1 Capacity 29, Cost 0

11. Circulation networks • Flow networks with b´ 0 are called circulation networks. • A feasible flow in a circulation network is called a feasible circulation.

12. Unknown Balances!! Unknown Balances!!

14. Integrality theorem for min cost flow If a flow network with costs has integral capacities and balances and a feasible flow in the network exists, then there is a minimum cost feasible flow which is integral on every arc. (shown later by “type checking”)

15. Assignment problem • Given integer weight matrix (w(i,j)), 1 · i,j ·n. • Find a permutationon {1,..,n} maximizing iw(i,(i)).

16. Min cost flow model of Ahuja et al • Ahuja operates with non-reduced flows, we work with reduced flows (net flows). They do not require flows and costs to be skew-symmetric.

17. Unreduced vs net flows -2 2 4 2 Net flow Unreduced flow

18. Min cost flow model of Ahuja • Ahuja operates with non-reduced flows, we work with reduced flows (net flows). They do not require flows and costs to be skew symmetric. • The difference matters only bidirectional arcs (an arc from u to v and an arc from v to u) with positive capcity in each direction. • One can translate (reduce) the Ahuja version to our version (exercise).

19. Tanker Scheduling Problem

20. All balances 0 Capacity 4, Cost 1 Capacity 1, Cost 0 Capacity 1, Cost 0 Capacity 1, Lower Bound 1, Cost 0

21. Hopping Airplane Problem • An airplane must travel from city 1, to city 2, to city 3, .., to city n. At most p passengers can be carried at any time. • bijpassengers want to go from city i to city j and will pay fij for the trip. • How many passengers should be picked up at each city in order to maximize profits?

23. Local Search Pattern LocalSearch(ProblemInstance x) y := feasible solution to x; while 9z∊N(y): v(z)<v(y)do y := z; od; returny; N(y) is a neighborhood of y.

24. Local search checklist Design: • How do we find the first feasible solution? • Neighborhood design? • Which neighbor to choose? Analysis: • Partial correctness? (termination )correctness) • Termination? • Complexity?

25. The first feasible flow? • Because of negative capacities and balance constraints, finding the first flow is non-trivial. • The zero flow may not work and there may not be anyfeasible flow for a given instance. • We can find the first feasible flow, if one exists, by reducing this problem to a max flow problem.

26. Neighborhood design • Given a feasible flow, how can we find a slightly different (and hopefully slightly better) flow?

27. All other costs 0, all balances 0. Capacity 29, Cost -1 Capacity 29, Cost 0

28. The residual network • Let G=(V,E,c,b,k) be a flow network with costs and let f be a flow in G. • The residual networkGf is the flow network with costs inherited from G and edges, capacities and balances given by: Ef = {(u,v) 2V£V| f(u,v) < c(u,v)} cf(u,v) = c(u,v) - f(u,v) ¸ 0 bf(u)=0

29. Lemma 3 Let • G=(V,E,c,b,k) be a flow network with costs • f be a feasible flow in G • Gfbe the residual network • f’ be a feasible flow in Gf Then • f+f’ is a feasible flow in G with cost(f+f’)=cost(f)+cost(f’)

30. Lemma 4 Let • G=(V,E,c,b,k) be a flow network with costs • f be a feasible flow in G • f’ be a feasible flow in G Then • f’-f is a feasible flow in Gf

31. Cycle Flows • Let C = (u1!u2!u3 … !uk=u1) be a simple cycle in G. • The cycle flowC is the circulation defined by C(ui, ui+1) =  C(ui+1, ui) = -  C(u,v) = 0 otherwise.

32. Augmenting cycles • Let G be a flow network with costs and Gfthe residual network. • An augmenting cycleC=(u1, u2, …, ur = u1) is a simple cycle in Gf for which cost(C)<0 where  is the minimum capacity cf(ui, ui+1), i = 1…r-1

33. Klein’s algorithm for min cost flow MinCostFlow(G) Using max flow algorithm, find feasible flow f in G (if no such flow exist, abort). while(9 augmenting cycle C in Gf){  = min{cf(e) | e on C} f := f + C } outputf

34. Klein’s algorithm • If Klein’s algorithm terminates it produces a feasible flow in G (by Lemma 3). • Is it partially correct? • Does it terminate?

35. Circulation Decomposition Lemma Let G be a circulation network with no negative capacities. Let f be a feasible circulation in G. Then, f may be written as a sum of cycle flows: f = C11 + C22 + … + Cmm where each cycle flow is a feasible circulation in G.

36. CDL ) Partial Correctness of Klein • Suppose f is not an minimum cost flow in G. We should show that Gfhas an augmenting cycle. • Let f * be a minimum cost flow in G. • f * - f is a feasible circulation in Gf of strictly negative cost (Lemma 4). • f * - f is a sum of cycle flows, feasible in Gf(by CDL). • At least one of them must have strictly negative cost. • The corresponding cycle is an augmenting cycle.

37. Termination • Assume integer capacities and balances. • For any feasible flow f occuring in Klein’s algorithm and any u,v, the flow f(u,v) is an integer between –c(v,u) and c(u,v). • Thus there are only finitely many possibilities for f. • In each iteration, f is improved – thus we never see an old f again. • Hence we terminate.

38. Integrality theorem for min cost flow If a flow network with costs has integral capacities and balances and a feasible flow in the network exists, then there is a minimum cost feasible flow which is integral on every arc. Proof by “type checking” Klein’s algorithm

39. Complexity • How fast can we perform a single iteration of the local search? • How many iterations do we have?

40. Complexity of a single iteration • An iteration is dominated by finding an augmenting cycle. • An augmenting cycle is a cycle (u1, u2, … ur=u1) in Gf with ik(ui,ui+1) < 0 • How to find one efficiently? Exercise 7.

41. Number of iterations • As Ford-Fulkerson, Klein’s algorithm may use an exponential number of iterations, if care is not taken choosing the augumentation (Exercise 6). • Fact:If the cycle with minimum average edge cost is chosen, there can be at most O(|E|2 |V| log |V|) iterations.

42. Generality of Languages Linear Programs Algorithm Simplex Algorithm, Interior Point Algorithms Reduction Klein’s algorithm Min Cost Flow Algorithm Reduction Decreasing efficiency Algorithm Max (s,t)-Flow Edmonds-Karp Increasing generality Reduction Maximum Cardinality Matching Algorithm Hopcroft-Karp (dADS)